Contact structures and open books
We explore the correspondence between open books and contact structures on three-manifolds. We begin with the necessary definitions and proofs for the correspondence; then we obtain technical results to understand the relationship between compatibility, Murasugi sum, and homotopy classes of plane fields. We use these to prove Harer’s conjecture on fibered links. Finally we explore the topological meaning of overtwistedness. We define sobering arcs in an open book and give a condition, then a criterion, for an open book to be overtwisted. Using this condition we give examples of overtwisted open books which come from positive configuration graphs. Finally we explore the limits of our sobering arc technique.